I encounter the problem that I would like to extend the implicit function theorem (for real numbers) to a global version.
The classical implicit function theorem is given by the following: Assume $F: \mathbb{R}^{n+m} \to \mathbb{R}^m$ is a continuously differentiable function and assume there is some $(x_0,y_0) \in \mathbb{R}^{n+m}$ such that $F(x_0,y_0) = 0$ and such that the Jacobian matrix (with respect to $y$) at $(x_0,y_0)$ is invertible. Then, there exists an open set $U \subseteq \mathbb{R}^n$ around $x_0$ and a unique continuously differentiable function $g: U \to \mathbb{R}^m$ such that $g(x_0) = y_0$ and $F(x,g(x)) = 0$ for all $x \in U$.
However, I would like to obtain a global version of the theorem which guarantees that $F(x,g(x)) = 0$ for all $x \in \mathbb{R}^n$ and that $g$ is unique on the whole space $\mathbb{R}^n$. Does anyone know easy conditions for this and/or a citeable reference such as a book or a published paper?
Thanks in advance!
The way you've set things up, it looks like you've set $n = m = 1$. In that case, the additional condition that there exists a $c > 0$ such that $\frac{\delta f}{\delta y}(x, y) \ge c$ for every $(x, y)$ in $\Bbb R^2$ is sufficient to get what you want. I don't know about general $n$ and $m$, however.
Source: Advanced Calculus 2E by Patrick M. Fitzpatrick
(I had to show this to be true as an exercise.)